Question

ABSTRACT REASONING A function is written in intercept form with a > 0\. What happens to the vertex of the graph as a increases? as a approaches 0?

Short answer

As a increases, the parabola stretches vertically but the x coordinate of the vertex remains unchanged. As a approaches 0, the parabola flattens and the vertex appears to move towards the x axis.

Step-by-step solution

Express The Vertex In Terms Of a, p, and q

In the expression of a parabola in the intercept form, the x-coordinate of the vertex can be found using the formula:\(x_{vertex}=\frac{p+q}{2}\). The y value of the vertex can also be obtained by substituting the x value of the vertex into the function. This results in \(y_{vertex}=a(\frac{p+q}{2}-p)(\frac{p+q}{2}-q)\). The vertex therefore is \((\frac{p+q}{2}, a(\frac{p+q}{2}-p)(\frac{p+q}{2}-q))\) and both x and y values do not depend on a.

Observe The Vertex As a Increases

As a increases, the functions stretches vertically but the x coordinate of the vertex remains the same because it does not depend on a. The minimum or maximum value (y coordinate of vertex) will scale linearly with a but the position along the x axis will stay same.

Observe The Vertex As a Approaches 0

As a approaches 0, the graph of the function becomes more flattened. However, since the vertex coordinates do not depend on a, the x coordinate of the vertex remains the same and the y coordinate goes toward zero. Hence the vertex will appear to move toward the x axis.

Key concepts

Vertex of a Parabola

The vertex is a key feature of any parabola, representing its peak or trough, depending on whether it opens upwards or downwards. In the intercept form of a quadratic function, the vertex has a precise mathematical definition. Here, the x-coordinate of the vertex is calculated as the average of the intercepts, given by the formula:

• \(x_{vertex} = \frac{p+q}{2}\)

This formula shows that the x-coordinate is entirely unaffected by changes in the parameter \(a\), as it solely depends on the values of \(p\) and \(q\).

The y-coordinate, however, is calculated by substituting this x value back into the quadratic function, which gives us:

• \(y_{vertex} = a\left(\frac{p+q}{2} - p\right)\left(\frac{p+q}{2} - q\right)\)

This means the height of the vertex (y-coordinate) is influenced by \(a\), changing as \(a\) changes. However, the horizontal position of the vertex remains steady, making the vertex's x-coordinate independent of \(a\).

Effects of Parameter Changes

In the algebraic study of quadratic functions, particularly in intercept form, it is vital to understand how changing parameters affects the graph. The parameter \(a\) plays a crucial role. When \(a\) is increased, the graph of the parabola becomes narrower, as if it is being stretched vertically. In this case, while the x-coordinate of the vertex remains the same, the y-coordinate increases in magnitude (absolute value goes up, but if \(a\) is negative, y would be more negative, falling lower if parabola is upside-down).

In contrast, as \(a\) approaches zero, the parabola flattens. Again, the x-coordinate stays untouched, but the y-coordinate of the vertex nears zero. This flattening effect makes the graph appear to get wider, smoothing out toward the x-axis. Knowing these effects helps in predicting and sketching the graph's shape without even plotting it.

Intercept Form of Quadratic Function

Understanding the intercept form of a quadratic function is crucial for dissecting how different elements influence the graph. In its intercept form, the quadratic function is expressed as:

• \(y = a(x - p)(x - q)\)

Here, \(p\) and \(q\) represent the x-intercepts or roots of the parabola. The parameter \(a\) dictates the direction and steepness of the parabola's opening.The intercept form makes it convenient to find the x-intercepts instantly. Importantly, this form allows the pinpointing of the x-coordinate of the vertex easily with \(x_{vertex} = \frac{p+q}{2}\). This algebraic property is useful not just in manual graphing but also in optimization problems, where the maximum or minimum points are crucial. Recognizing these characteristics allows students to intuitively grasp the geometric and algebraic significance of each parameter.